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Thread: Incorporating dark matter/energy into Einstein's field equations

  1. #1 Incorporating dark matter/energy into Einstein's field equations 
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    Just as an informative note :

    I came across this article, and the corresponding paper from 2012:

    GRAVITATIONAL FIELD EQUATIONS AND
    THEORY OF DARK MATTER AND DARK ENERGY

    (TIAN MA AND SHOUHONG WANG)

    "A pair of mathematicians -- one from Indiana University and the other from Sichuan University in China -- have proposed a unified theory of dark matter and dark energy that alters Einstein's equations describing the fundamentals of gravity"


    Link article :

    http://www.sciencedaily.com/releases/2012/09/120906092059.htm

    Link paper :

    http://arxiv.org/pdf/1206.5078.pdf


    Last edited by Noa Drake; May 16th, 2014 at 06:28 AM.
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    Moderator Moderator Markus Hanke's Avatar
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    Quote Originally Posted by Noa Drake View Post
    (can a moderator perhaps correct the title please > m to p, thank you)
    Done.

    Interesting paper. Unfortunately I am not very familiar with Sobolev spaces, so much of the maths in the derivation is a little beyond me. In any case, the basic idea is simple - abandon the requirement that the energy-momentum tensor must be divergence-free, and instead introduce a scalar field that contributes as a source of energy-momentum :



    This of course means that the vacuum is no longer Ricci flat, so the scalar field plays a role similar to the cosmological constant, except that it is allowed to vary across space-time. The coupling between the scalar field and the energy-momentum tensor allows us to unify DM and DE into a single source.

    The hypothesis is interesting, it's just that to make it work we would need a new scalar field as well as a fifth fundamental force to be present across the universe; so, we would be abandoning two hypothetical entities only to replace them with two new hypothetical entities. To be perfectly honest, as interesting as the paper is, I somehow fail to see the basic point since we don't really gain anything by doing this.


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    Thank you Markus.

    Indeed no direct call for Eureka here.

    But the interesting part to me is this :
    "The second-order covariant derivative would be the geometric analog of a second order derivative in calculus which measures how the rate of change of a quantity is itself changing."


    or as you stated :

    "
    so the scalar field plays a role similar to the cosmological constant, except that it is allowed to vary across space-time"


    Can this concept of local variations then be interpreted as the cause of gravity ?


    As for the gain : an incorporation of dark matter and dark energy in a theory that includes gravity.

    I'm not sure, but are they not aiming to install this as the cause of the fundamental forces, instead of being a fifth force ?
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    Moderator Moderator Markus Hanke's Avatar
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    Quote Originally Posted by Noa Drake View Post
    "The second-order covariant derivative would be the geometric analog of a second order derivative in calculus which measures how the rate of change of a quantity is itself changing."
    That's the case anyway. The covariant derivative is just a generalisation of the ordinary derivative to arbitrary manifolds.

    Can this concept of local variations then be interpreted as the cause of gravity
    No, it has nothing to do with it. What the paper describes is basically an extension of General Relativity to include an additional scalar field as well. The cause of gravity is still geodesic deviation, just as in standard GR, only the numbers are different now.

    As for the gain : an incorporation of dark matter and dark energy in a theory that includes gravity.
    True, but the price you pay for that gain is the introduction of two new completely hypothetical entities.

    I'm not sure, but are they not aiming to install this as the cause of the fundamental forces, instead of being a fifth force ?
    No, they postulate a specific form of coupling of the scalar field to the energy-momentum tensor field; this is equivalent to a new fundamental interaction. If you look at the conclusions paragraph on page 31, the four main results are nicely summarised there.

    The paper as a whole presents a model that I would consider physically viable - so far as I can see it is in accord with empirical data, and it is based on sound maths and sound physics, so no objections here. All I was trying to point out is that the presence of the scalar field as well as its coupling to sources of energy-momentum are merely postulates, so we don't really eliminate any hypothetical entities from our understanding, we only shift a hypothesis from one proposal ( DM and DE ) to another ( the scalar field in this instance ).

    Am I making sense ?
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    Ok, thank you.

    Looking further at the conclusions in the paper :

    They start from a scalar potential in the universe that would be zero if the universe were homogenous.
    As we observe that matter is distributed diversely over the universe, this leads then to negative and positive areas expressed as local scalar potential energy density, right ?

    > Does this not indicate an overlap of causes ? I mean the coming into existence of spacetime around a planet (area of intense mass concentration compared to other regions of the universe) for instance, is then the same thing as saying that the local scalar potential energy density is negative there, is the same as saying that dark matter is spacetime is negative scalar potential energy, no ?

    I am not saying it is that way, i am just contemplating on the consequenses of their approach.
    As if they are placing spacetime in a larger context.

    (Needless to say that i am in favor of the hypothesis of a once homogenous universe that became diverse starting from the same building blocks..)
    Last edited by Noa Drake; May 16th, 2014 at 07:59 AM.
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    I was not at first intending to provide personal oppinions on this, but one thing leads to another, and a forum is a place for discussion,
    i would not object to a move to another section.
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    Quote Originally Posted by Noa Drake View Post
    As we observe that matter is distributed diversely over the universe, this leads then to negative and positive areas expressed as local scalar potential energy density, right ?
    Yes, this is essentially how I understand their proposal. The scalar field is associated with an energy density which has a positive and a negative component, the sum total of which varies over space and time.

    > Does this not indicate an overlap of causes ? I mean the coming into existence of spacetime around a planet (area of intense mass concentration compared to onther regions of the universe) for instance, is then the same thing as saying that the local scalar potential energy density is negative there, is the same as saying that dark matter is spacetime is negative scalar potential energy, no ?
    I'm afraid I am not sure what you are trying to say here. The basic idea is to abandon the concepts of DM and DE, and model their effects directly via changes in the geometry of space-time, caused by the presence of the scalar field and its associated energy density.
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    Quote Originally Posted by Noa Drake View Post
    (Needless to say that i am in favor of the hypothesis of a once homogenous universe that became diverse starting from the same building blocks..)
    This seems to me more or less exactly what standard cosmology suggests - you start with a hot, dense, very homogeneous and uniform state, and let it cool and expand into what we see today.
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    "changes in the geometry of space-time, caused by the presence of the scalar field and its associated energy density"

    Indeed, but the step in between these 2 is the diverse distribution of matter over the universe, which causes the values of the scalar field.

    That is why i assumed that this system is fundamental for gravity, because it is at the source of spacetime geometry.
    A scalar field means essentially attributing local values to each point, no ?

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    Quote Originally Posted by Noa Drake View Post
    A scalar field means essentially attributing local values to each point, no ?
    Correct, you associate one scalar value with each point.

    That is why i assumed that this system is fundamental for gravity, because it is at the source of spacetime geometry.
    The scalar field gives one contribution to the overall geometry, and the energy-momentum tensor gives another. If we were to make the scalar field everywhere zero, these field equations reduce to standard GR - in that sense it is a generalisation of the standard theory. The one major difference is that in vaccum ( ) the Einstein tensor does not vanish in this model, so the geometry of the vacuum is different than in standard GR - this is the mechanism through which the effects of DE and DM are modelled.

    To put it simply - the presence of the scalar field modifies the existing geometry. If you take away the scalar field, you still have space-time and its geometry, in the form of standard GR, it's just that the two geometries aren't the same.
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    Ok, that is clear.
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